Finite Group Databasebeta

← #34442913

Subgroups — #34442913

Order 20224.

546 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773912C211yesmaximal, characteristic, complemented
2#23772171C211yesmaximal, characteristic
3#23765567C211yesmaximal, characteristic, complemented
4#23765563C211yesmaximal, characteristic, complemented
5#23765560C211yesmaximal, characteristic, complemented
6#23765556C211yesmaximal, characteristic, complemented
7#23765551C211yesmaximal, characteristic, complemented
8#23765546C211yesmaximal, characteristic, complemented
9#23765541C211yesmaximal, characteristic, complemented
10#23765536C211yesmaximal, characteristic, complemented
11#23765182C211yesmaximal, characteristic, complemented
12C2×Dic79⋊C16C211yesmaximal, characteristic, complemented
13#23765108C211yesmaximal, characteristic
14#23765105C211yesmaximal, characteristic
15C79×#2158C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17#19099116—21no—
18#19099031—21no—
19C22×C79⋊C16C2211yescharacteristic
20#19098967C2211yescharacteristic
21#19098967C2211yescharacteristic
22#19098964C2211yescharacteristic
23#19098964C2211yescharacteristic
24#19098963—21no—
25#19098960—21no—
26#19098657C2211yescharacteristic
27#19098657C2211yescharacteristic
28#19098650—21no—
29#19098650—21no—
30#19098645C2211yescharacteristic
31#19098645C2211yescharacteristic
32#19098601C2211yescharacteristic
33#19098599—21no—
34#19098557—21no—
35C2×C8×Dic79C2211yescharacteristic
36#19098390C2211yescharacteristic
37#19098390C2211yescharacteristic
38#19098388C2211yescharacteristic
39#19098388C2211yescharacteristic
40Dic79⋊C16C2211yescharacteristic
41Dic79⋊C16C2211yescharacteristic
42Dic79⋊C16C2211yescharacteristic
43Dic79⋊C16C2211yescharacteristic
44#19098368C2211yescharacteristic
45#19098368C2211yescharacteristic
46#19098366C2211yescharacteristic
47#19098366C2211yescharacteristic
48#19098365C2211yescharacteristic
49#19098365C2211yescharacteristic
50#19098363C2211yescharacteristic
51#19098363C2211yescharacteristic
52C22×C16×C79C2211yescharacteristic
53#427×C79C2211yescharacteristic
54#427×C79C2211yescharacteristic
55#367×C79C2211yescharacteristic
56#367×C79C2211yescharacteristic
57#366×C79C2211yescharacteristic
58#366×C79C2211yescharacteristic
59C2×#11929624—21no—
60#14579183—22no—
61#14579183Q814yescomplemented
62C2×Dic316C2311yescharacteristic
63C2×#11929613—21no—
64C2×D632C2311yescharacteristic
65C22×C79⋊C8D411yescharacteristic
66#14579160—21no—
67#14579160—21no—
68#14579152—21no—
69#14579152—21no—
70#14579138—41no—
71#14579138—41no—
72C4×Dic158—41no—
73C4×Dic158—41no—
74#14579126—21no—
75#14579126—21no—
76#14579123—21no—
77#14579123—21no—
78(C22⋊D79)⋊C4—41no—
79(C22⋊D79)⋊C4—41no—
80#14579120—21no—
81#14579118—41no—
82C2×C4×Dic79D411yescharacteristic
83C42×D79—41no—
84C2×C79⋊C16—22no—
85C2×C79⋊C16C2311yescharacteristic
86C2×C79⋊C16C2311yescharacteristic
87#14579095C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579094—21no—
90#14579094—21no—
91#14579093—21no—
92#14579092—21no—
93#14579092—21no—
94#14579084—21no—
95#14579084—21no—
96#14579081—21no—
97#14579081—21no—
98C8×Dic79D411yescharacteristic
99C8×Dic79D411yescharacteristic
100C8×Dic79C2311yescharacteristic
101C8×Dic79C2311yescharacteristic
102C22×C8×C79C2311yescharacteristic
103#117×C79—21no—
104#117×C79—21no—
105C2×C16×C79—22no—
106C2×C16×C79C2311yescharacteristic
107C2×C16×C79C2311yescharacteristic
108#107×C79C2311yescharacteristic
109#107×C79C2311yescharacteristic
110#107×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
112#106×C79—21no—
113#106×C79—21no—
114#11929624—24no—
115#11929624—42no—
116C2×Dic158—21no—
117C2×Dic158—41no—
118C2×Dic158—41no—
119C2×D316—21no—
120C2×C22⋊D79—41no—
121C22×Dic79D811yescharacteristic
122C22×Dic79D811yescharacteristic
123C2×C4×D79—41no—
124C2×C4×D79—41no—
125C2×C4×D79—41no—
126Dic316—21no—
127Dic316C2×Q812yes—
128#11929613—24no—
129D632—21no—
130D632C2×Q812yes—
131C2×C79⋊C8—22no—
132C2×C79⋊C8C2×D411yescharacteristic
133C2×C79⋊C8C2×D411yescharacteristic
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—41no—
137C4⋊Dic79—41no—
138Dic79⋊C4—41no—
139Dic79⋊C4—41no—
140Dic79⋊C4—41no—
141Dic79⋊C4—41no—
142Dic79⋊C4—41no—
143Dic79⋊C4—41no—
144C22⋊Dic79—41no—
145(C2×D79)⋊C4—41no—
146(C2×D79)⋊C4—41no—
147C4×Dic79—41no—
148C4×Dic79—41no—
149C4×Dic79—41no—
150C4×Dic79D811yescharacteristic
151C4×Dic79D811yescharacteristic
152C4×Dic79C2×D411yescharacteristic
153C4×Dic79C2×D411yescharacteristic
154C79⋊C16—21no—
155C79⋊C16—21no—
156C22×C4×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2×Q812yes—
159C2×C8×C79#4112yes—
160C2×C8×C79C2411yescharacteristic
161C4⋊C4×C79—21no—
162C4⋊C4×C79—21no—
163C4⋊C4×C79—21no—
164C4⋊C4×C79—21no—
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C42×C79—41no—
168C42×C79—41no—
169C16×C79—21no—
170C16×C79—21no—
171C23×C79C2×D811yescharacteristic
172C22×D79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79C22⋊Q812yes—
178C2×C4×C79C2×D811yescharacteristic
179C2×C4×C79#13712yes—
180C2×C4×C79C22×D411yescharacteristic
181C22⋊D79—44no—
182C2×Dic79—22no—
183C2×Dic79—41no—
184C2×Dic79—41no—
185C2×Dic79—41no—
186C2×Dic79—41no—
187C2×Dic79C2×D811yescharacteristic
188C2×Dic79C2×D811yescharacteristic
189C2×Dic79C2×D811yescharacteristic
190C2×Dic79C2×D811yescharacteristic
191D316—22no—
192D316—41no—
193C4×D79—44no—
194C4×D79—82no—
195C4×D79—82no—
196Dic158—22no—
197Dic158—41no—
198Dic158—82no—
199Dic158—82no—
200C8×C79C22⋊Q812yes—
201C8×C79C22×Q811yescharacteristic
202C8×C79C2×#4111yescharacteristic
203C79⋊C8—21no—
204C79⋊C8—21no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79#47412yes—
208C22×C79C22×D811yescharacteristic
209C2×D79—42no—
210C2×D79—81no—
211C2×D79—81no—
212C4×C79—81no—
213C4×C79—81no—
214C4×C79—81no—
215C4×C79—81no—
216C4×C79#47412yes—
217C4×C79C2×C22⋊Q811yescharacteristic
218C4×C79C2×#13711yescharacteristic
219Dic79—21no—
220Dic79—21no—
221Dic79—21no—
222Dic79—42no—
223Dic79—81no—
224Dic79—81no—
225Dic79#50812yes—
226#33626—791nomaximal, Sylow
227C2×C79—22no—
228C2×C79C2×#47411yescharacteristic
229C2×C79C2×#50811yescharacteristic
230C2×C79#341411yescharacteristic
231D79—82no—
232#3160—791no—
233C4×#135—791no—
234#2229—791no—
235#2229—791no—
236#2228—791no—
237#2228—791no—
238#2227—791no—
239#2226—791no—
240#2225—791no—
241#2224—791no—
242#2158—791no—
243#2158D7911yescharacteristic, complemented
244C2×C4⋊C16—791no—
245#2138—791no—
246#2138—791no—
247C79#3362611yescharacteristic, complemented, Sylow
248C2×#135—791no—
249#533—1581no—
250C4×#41—1581no—
251C22×C16—791no—
252C22×C16C2×D7911yescharacteristic
253#500—791no—
254#500—791no—
255#499—791no—
256#499—791no—
257#499—1581no—
258#498—1581no—
259C4×Q16—791no—
260C4×Q16—791no—
261C4×SD16—1581no—
262C4×SD16—1581no—
263C4×D8—791no—
264C4×D8—791no—
265#427—791no—
266#427—1581no—
267#427C2×D7911yescharacteristic
268#427C2×D7911yescharacteristic
269#416—1581no—
270C2×C4×C8—791no—
271#367—791no—
272#367—791no—
273#367C2×D7911yescharacteristic
274#367C2×D7911yescharacteristic
275#366—791no—
276#366—791no—
277#366C2×D7911yescharacteristic
278#366C2×D7911yescharacteristic
279C4⋊C16—791no—
280C4⋊C16—791no—
281C4⋊C16—791no—
282C4⋊C16—791no—
283#358—791no—
284#358—791no—
285#358—791no—
286#358—791no—
287#357—791no—
288#357—791no—
289#357—791no—
290#357—791no—
291C2×#41—1581no—
292#135—794no—
293#135—1582no—
294C2×Q16—791no—
295C2×SD16—1581no—
296C2×D8—791no—
297C22×C8—791no—
298C22×C8C22×D7911yescharacteristic
299C4⋊Q8—1581no—
300C4⋊Q8—1581no—
301#125—1581no—
302#125—1581no—
303C22⋊Q8—3161no—
304C22⋊Q8—3161no—
305C4×Q8—3161no—
306C4×Q8—3161no—
307C4×Q8—1581no—
308C4×Q8—1581no—
309C4×D4—3161no—
310C4×D4—3161no—
311C4×D4—1581no—
312C4×D4—1581no—
313#117—21no—
314#117—21no—
315#117—3161no—
316#117—1581no—
317C2×C42—3161no—
318C2×C42—791no—
319C2×C16—22no—
320C2×C16—791no—
321C2×C16—791no—
322C2×C16—1582no—
323C2×C16C22×D7911yescharacteristic
324C2×C16C22×D7911yescharacteristic
325#107—791no—
326#107—791no—
327#107—1581no—
328#107—1581no—
329#107C22×D7911yescharacteristic
330#107C22×D7911yescharacteristic
331#107C22×D7911yescharacteristic
332#107C22×D7911yescharacteristic
333#106—21no—
334#106—21no—
335#106—1581no—
336#106—1581no—
337#106—1581no—
338#103—1581no—
339#103—1581no—
340#102—1581no—
341#102—1581no—
342C4×C8—791no—
343C4×C8—791no—
344C4×C8—791no—
345C4×C8—791no—
346#41—3162no—
347#41—1584no—
348C2×Q8—3161no—
349C2×Q8—3161no—
350C2×Q8—1581no—
351C2×D4—3161no—
352C2×D4—1581no—
353C22×C4—3161no—
354C22×C4—3161no—
355C22×C4—3161no—
356C22×C4—791no—
357C22×C4—791no—
358C22×C4D4×D7911yescharacteristic
359Q16—792no—
360Q16—1581no—
361SD16—1584no—
362D8—792no—
363D8—1581no—
364C2×C8—791no—
365C2×C8—791no—
366C2×C8—1582no—
367C2×C8D4×D7911yescharacteristic
368C2×C8Q8×D7912yes—
369C2×C8#1192962412yes—
370C2×C8C23×D7911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—3161no—
376C4⋊C4—3161no—
377C4⋊C4—3161no—
378C4⋊C4—3161no—
379C4⋊C4—3161no—
380C4⋊C4—3161no—
381C4⋊C4—3161no—
382C4⋊C4—3161no—
383C4⋊C4—1581no—
384C4⋊C4—1581no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—3161no—
388C22⋊C4—3161no—
389C22⋊C4—3161no—
390C42—41no—
391C42—41no—
392C42—3161no—
393C42—3161no—
394C42—3161no—
395C42—791no—
396C42—791no—
397C42—791no—
398C42—791no—
399C16—21no—
400C16—21no—
401C16—1581no—
402C16—1581no—
403C23—3161no—
404C23D8×D7911yescharacteristic
405Q8—3161no—
406Q8—6322no—
407Q8—6322no—
408Q8—1582no—
409D4—3164no—
410D4—3161no—
411D4—1582no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—3161no—
417C2×C4—3164no—
418C2×C4—3161no—
419C2×C4—3161no—
420C2×C4—3161no—
421C2×C4—791no—
422C2×C4—791no—
423C2×C4—791no—
424C2×C4—791no—
425C2×C4—6322no—
426C2×C4—6322no—
427C2×C4—1582no—
428C2×C4#1457913912yes—
429C2×C4D8×D7911yescharacteristic
430C2×C4#1457919312yes—
431C2×C4C2×D4×D7911yescharacteristic
432C8—1581no—
433C8—1581no—
434C8#1457913912yes—
435C8C2×Q8×D7911yescharacteristic
436C8C2×#1192962411yescharacteristic
437C22—21no—
438C22—21no—
439C22—3162no—
440C22—6321no—
441C22—6321no—
442C22#1909884212yes—
443C22C2×D8×D7911yescharacteristic
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—3162no—
449C4—792no—
450C4—6321no—
451C4—6321no—
452C4—1581no—
453C4—1581no—
454C4—1581no—
455C4#1909884212yes—
456C4C2×#1457913911yescharacteristic
457C4C2×#1457919311yescharacteristic
458C2—22no—
459C2—6322no—
460C2C2×#1909884211yescharacteristic
461C2#508×D7911yescharacteristic
462C2#2377653311yescharacteristic
subgroupG/H
centerC2×C4D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2158C2
socleC22×C79C22×D8
prime radicalC2×#14579183C22
hypercenter#2158D79
hypocenterC79#33626
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23773912C2semidirect
3. #23765567C2semidirect
4. #23765563C2semidirect
5. #23765560C2semidirect
6. #23765556C2semidirect
7. #23765551C2semidirect
8. #23765546C2semidirect
9. #23765541C2semidirect
10. #23765536C2semidirect
11. #23765182C2semidirect
12. C2×Dic79⋊C16C2semidirect
15. C79×#2158C2semidirect
61. #14579183Q8semidirect
243. #2158D79semidirect
247. C79#33626semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237739122
#237721712
#237655672
#237655632
#237655602
#237655562
#237655512
#237655462
#237655412
#237655362
#237651822
C2×Dic79⋊C162
#237651082
#237651052
C79×#21582
#3362679